Assume that * is an operation on S with identity element e and that,
x * (y * z) = (x * z) * y
for all x,y,z [itex]\in[/itex] S. Prove that * is commutative and associative.
2. The attempt at a solution
I want to prove commutativity first as that may make it easier to prove associativity. I have made a few Cayley tables proving this, but I am not sure if that is a real proof. For example, having x*z = e, y*z = x, and x*x = y. These are arbitrary selections as long as no two elements are in the same row or column (and LHS = RHS). After that, I can fill in the table one and only one way regardless of my above selections within those two parameters. Here is a picture of what I mean:
*|e x y z
e|e x y z
x|x y _ _
y|y _ _ _
z|z e x _
Filling in the rest of the table, I will get symmetry along the diagonal implying that * is commutative (and associative also).
However, I want to prove this without a table using the identity given. My brain is kind of tired right now and I feel it is very obvious. But for commutativity, I must get rid of one of the elements since commutativity is only between two elements. Can I assume, as I did with the table, that an arbitrary selection of two elements will equal the identity element, e.g. x*z = e? Even if I do, I can't seem to 'remove' an element... I guess I need a break. Thanks in advance.